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A126999
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Zero-one fractional-part array for the golden ratio; a rectangular array T by antidiagonals.
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1
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1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| (Row 1) = (Column 1) = A005614 (infinite Fibonacci word).
(Row 2) = (Column 2) = A123740
(Row 3) = (Column 3) = A187944
(Row 4) = (Column 4) = A187950
(Main Diagonal) = A078588
See A187950 for connections to left-shifted sums of the infinite Fibonacci word.
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FORMULA
| T(n,k)={nr}+{kr}-{nr+kr}, where r=(1+sqrt(5))/2 and { } denotes fractional part;, i.e., {x}=x-Floor(x).
T(n,k)=[nr]+[kr]-[nr+kr], where []=floor.
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EXAMPLE
| Northwest corner:
1 0 1 1 0 1 0 1 1
0 0 1 0 0 0 0 1 0
1 1 1 1 0 1 1 1 1
1 0 1 0 0 1 0 1 1
0 0 0 0 0 0 0 1 0
1 0 1 1 0 1 1 1 1
T(3,3)=1 because 2{3x}-{6x}=1.
The antidiagonals form a triangle with these first six rows:
1
0 0
1 0 1
1 1 1 1
0 0 1 0 0
1 0 1 1 0 1
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MATHEMATICA
| r=(1+5^(1/2))/2;
T[k_, n_]:=Floor[n*r+k*r]-Floor[n*r]-Floor[k*r]
TableForm[Table[T[n, k], {k, 1, 20}, {n, 1, 20}]]
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CROSSREFS
| Cf. A005614, A123740, A078588, A126700, A126701, A188294, A187950.
Sequence in context: A188578 A104105 A143221 * A120527 A071004 A102560
Adjacent sequences: A126996 A126997 A126998 * A127000 A127001 A127002
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KEYWORD
| nonn,tabl
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu), Jan 01 2007
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